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<div class="header">
  <div class="summary">
<a href="#var-members">Variables</a>  </div>
  <div class="headertitle">
<div class="title">Ring<div class="ingroups"><a class="el" href="group__group-concepts.html">Concepts</a></div></div>  </div>
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<a name="details" id="details"></a><h2 class="groupheader">Description</h2>
<p>The <code>Ring</code> concept represents <code>Group</code>s that also form a <code>Monoid</code> under a second binary operation that distributes over the first. </p>
<p>A <a href="http://en.wikipedia.org/wiki/Ring_(mathematics)">Ring</a> is an algebraic structure built on top of a <code>Group</code> which requires a monoidal structure with respect to a second binary operation. This second binary operation must distribute over the first one. Specifically, a <code>Ring</code> is a triple <code>(S, +, *)</code> such that <code>(S, +)</code> is a <code>Group</code>, <code>(S, *)</code> is a <code>Monoid</code> and <code>*</code> distributes over <code>+</code>, i.e. </p><div class="fragment"><div class="line">x * (y + z) == (x * y) + (x * z)</div>
</div><!-- fragment --><p>The second binary operation is often written <code>*</code> with its identity written <code>1</code>, in reference to the <code>Ring</code> of integers under multiplication. The method names used here refer to this exact ring.</p>
<h2><a class="anchor" id="autotoc_md160"></a>
Minimal complete definintion</h2>
<p><code>one</code> and <code>mult</code> satisfying the laws</p>
<h2><a class="anchor" id="autotoc_md161"></a>
Laws</h2>
<p>For all objects <code>x</code>, <code>y</code>, <code>z</code> of a <code>Ring</code> <code>R</code>, the following laws must be satisfied: </p><div class="fragment"><div class="line"><a class="code" href="group__group-_ring.html#ga052d31c269a6a438cc8004c9ad1efdfa">mult</a>(x, <a class="code" href="group__group-_ring.html#ga052d31c269a6a438cc8004c9ad1efdfa">mult</a>(y, z)) == <a class="code" href="group__group-_ring.html#ga052d31c269a6a438cc8004c9ad1efdfa">mult</a>(<a class="code" href="group__group-_ring.html#ga052d31c269a6a438cc8004c9ad1efdfa">mult</a>(x, y), z)          <span class="comment">// associativity</span></div>
<div class="line"><a class="code" href="group__group-_ring.html#ga052d31c269a6a438cc8004c9ad1efdfa">mult</a>(x, one&lt;R&gt;()) == x                              <span class="comment">// right identity</span></div>
<div class="line"><a class="code" href="group__group-_ring.html#ga052d31c269a6a438cc8004c9ad1efdfa">mult</a>(one&lt;R&gt;(), x) == x                              <span class="comment">// left identity</span></div>
<div class="line"><a class="code" href="group__group-_ring.html#ga052d31c269a6a438cc8004c9ad1efdfa">mult</a>(x, <a class="code" href="group__group-_monoid.html#gaeb5d4a1e967e319712f9e4791948896c">plus</a>(y, z)) == <a class="code" href="group__group-_monoid.html#gaeb5d4a1e967e319712f9e4791948896c">plus</a>(<a class="code" href="group__group-_ring.html#ga052d31c269a6a438cc8004c9ad1efdfa">mult</a>(x, y), <a class="code" href="group__group-_ring.html#ga052d31c269a6a438cc8004c9ad1efdfa">mult</a>(x, z)) <span class="comment">// distributivity</span></div>
<div class="ttc" id="agroup__group-_monoid_html_gaeb5d4a1e967e319712f9e4791948896c"><div class="ttname"><a href="group__group-_monoid.html#gaeb5d4a1e967e319712f9e4791948896c">boost::hana::plus</a></div><div class="ttdeci">constexpr auto plus</div><div class="ttdoc">Associative binary operation on a Monoid.</div><div class="ttdef"><b>Definition:</b> plus.hpp:47</div></div>
<div class="ttc" id="agroup__group-_ring_html_ga052d31c269a6a438cc8004c9ad1efdfa"><div class="ttname"><a href="group__group-_ring.html#ga052d31c269a6a438cc8004c9ad1efdfa">boost::hana::mult</a></div><div class="ttdeci">constexpr auto mult</div><div class="ttdoc">Associative operation of a Ring.</div><div class="ttdef"><b>Definition:</b> mult.hpp:47</div></div>
</div><!-- fragment --><h2><a class="anchor" id="autotoc_md162"></a>
Refined concepts</h2>
<p><code>Monoid</code>, <code>Group</code></p>
<h2><a class="anchor" id="autotoc_md163"></a>
Concrete models</h2>
<p><code><a class="el" href="structboost_1_1hana_1_1integral__constant.html" title="Compile-time value of an integral type.">hana::integral_constant</a></code></p>
<h2><a class="anchor" id="autotoc_md164"></a>
Free model for non-boolean arithmetic data types</h2>
<p>A data type <code>T</code> is arithmetic if <code>std::is_arithmetic&lt;T&gt;::value</code> is true. For a non-boolean arithmetic data type <code>T</code>, a model of <code>Ring</code> is automatically defined by using the provided <code>Group</code> model and setting </p><div class="fragment"><div class="line"><a class="code" href="group__group-_ring.html#ga052d31c269a6a438cc8004c9ad1efdfa">mult</a>(x, y) = (x * y)</div>
<div class="line">one&lt;T&gt;() = <span class="keyword">static_cast&lt;</span>T<span class="keyword">&gt;</span>(1)</div>
</div><!-- fragment --><dl class="section note"><dt>Note</dt><dd>The rationale for not providing a Ring model for <code>bool</code> is the same as for not providing Monoid and Group models.</dd></dl>
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Structure-preserving functions</h2>
<p>Let <code>A</code> and <code>B</code> be two <code>Ring</code>s. A function <code>f : A -&gt; B</code> is said to be a <a href="http://en.wikipedia.org/wiki/Ring_homomorphism">Ring morphism</a> if it preserves the ring structure between <code>A</code> and <code>B</code>. Rigorously, for all objects <code>x, y</code> of data type <code>A</code>, </p><div class="fragment"><div class="line">f(<a class="code" href="group__group-_monoid.html#gaeb5d4a1e967e319712f9e4791948896c">plus</a>(x, y)) == <a class="code" href="group__group-_monoid.html#gaeb5d4a1e967e319712f9e4791948896c">plus</a>(f(x), f(y))</div>
<div class="line">f(<a class="code" href="group__group-_ring.html#ga052d31c269a6a438cc8004c9ad1efdfa">mult</a>(x, y)) == <a class="code" href="group__group-_ring.html#ga052d31c269a6a438cc8004c9ad1efdfa">mult</a>(f(x), f(y))</div>
<div class="line">f(one&lt;A&gt;()) == one&lt;B&gt;()</div>
</div><!-- fragment --><p> Because of the <code>Ring</code> structure, it is easy to prove that the following will then also be satisfied: </p><div class="fragment"><div class="line">f(zero&lt;A&gt;()) == zero&lt;B&gt;()</div>
<div class="line">f(<a class="code" href="group__group-_group.html#ga02e81002f40ba52eac4cf1974c7e0cdb">negate</a>(x)) == <a class="code" href="group__group-_group.html#ga02e81002f40ba52eac4cf1974c7e0cdb">negate</a>(f(x))</div>
<div class="ttc" id="agroup__group-_group_html_ga02e81002f40ba52eac4cf1974c7e0cdb"><div class="ttname"><a href="group__group-_group.html#ga02e81002f40ba52eac4cf1974c7e0cdb">boost::hana::negate</a></div><div class="ttdeci">constexpr auto negate</div><div class="ttdoc">Return the inverse of an element of a group.</div><div class="ttdef"><b>Definition:</b> negate.hpp:26</div></div>
</div><!-- fragment --><p> which is to say that <code>f</code> will then also be a <code>Group</code> morphism. Functions with these properties interact nicely with <code>Ring</code>s, which is why they are given such a special treatment. </p>
<table class="memberdecls">
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Variables</h2></td></tr>
<tr class="memitem:ga052d31c269a6a438cc8004c9ad1efdfa"><td class="memItemLeft" align="right" valign="top">constexpr auto&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="group__group-_ring.html#ga052d31c269a6a438cc8004c9ad1efdfa">boost::hana::mult</a></td></tr>
<tr class="memdesc:ga052d31c269a6a438cc8004c9ad1efdfa"><td class="mdescLeft">&#160;</td><td class="mdescRight">Associative operation of a <code>Ring</code>.  <a href="group__group-_ring.html#ga052d31c269a6a438cc8004c9ad1efdfa">More...</a><br /></td></tr>
<tr class="separator:ga052d31c269a6a438cc8004c9ad1efdfa"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:gadea531feb3b0a1c5c3d777f7ab45e932"><td class="memTemplParams" colspan="2">template&lt;typename R &gt; </td></tr>
<tr class="memitem:gadea531feb3b0a1c5c3d777f7ab45e932"><td class="memTemplItemLeft" align="right" valign="top">constexpr auto&#160;</td><td class="memTemplItemRight" valign="bottom"><a class="el" href="group__group-_ring.html#gadea531feb3b0a1c5c3d777f7ab45e932">boost::hana::one</a></td></tr>
<tr class="memdesc:gadea531feb3b0a1c5c3d777f7ab45e932"><td class="mdescLeft">&#160;</td><td class="mdescRight">Identity of the <code>Ring</code> multiplication.  <a href="group__group-_ring.html#gadea531feb3b0a1c5c3d777f7ab45e932">More...</a><br /></td></tr>
<tr class="separator:gadea531feb3b0a1c5c3d777f7ab45e932"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:ga0ee3cff9ec646bcc7217f00ee6099b72"><td class="memItemLeft" align="right" valign="top">constexpr auto&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="group__group-_ring.html#ga0ee3cff9ec646bcc7217f00ee6099b72">boost::hana::power</a></td></tr>
<tr class="memdesc:ga0ee3cff9ec646bcc7217f00ee6099b72"><td class="mdescLeft">&#160;</td><td class="mdescRight">Elevate a ring element to its <code>n</code>th power.  <a href="group__group-_ring.html#ga0ee3cff9ec646bcc7217f00ee6099b72">More...</a><br /></td></tr>
<tr class="separator:ga0ee3cff9ec646bcc7217f00ee6099b72"><td class="memSeparator" colspan="2">&#160;</td></tr>
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<h2 class="groupheader">Variable Documentation</h2>
<a id="ga052d31c269a6a438cc8004c9ad1efdfa"></a>
<h2 class="memtitle"><span class="permalink"><a href="#ga052d31c269a6a438cc8004c9ad1efdfa">&#9670;&nbsp;</a></span>mult</h2>

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          <td class="memname">constexpr auto boost::hana::mult</td>
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<p><code>#include &lt;<a class="el" href="fwd_2mult_8hpp.html">boost/hana/fwd/mult.hpp</a>&gt;</code></p>
<b>Initial value:</b><div class="fragment"><div class="line">= [](<span class="keyword">auto</span>&amp;&amp; x, <span class="keyword">auto</span>&amp;&amp; y) -&gt; decltype(<span class="keyword">auto</span>) {</div>
<div class="line">        <span class="keywordflow">return</span> tag-dispatched;</div>
<div class="line">    }</div>
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<p>Associative operation of a <code>Ring</code>. </p>
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">x,y</td><td>Two <code>Ring</code> elements to combine with the <code>Ring</code> binary operation.</td></tr>
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</dl>
<h2><a class="anchor" id="autotoc_md286"></a>
Cross-type version of the method</h2>
<p>The <code>mult</code> method is "overloaded" to handle distinct data types with certain properties. Specifically, <code>mult</code> is defined for <em>distinct</em> data types <code>A</code> and <code>B</code> such that</p><ol type="1">
<li><code>A</code> and <code>B</code> share a common data type <code>C</code>, as determined by the <code>common</code> metafunction</li>
<li><code>A</code>, <code>B</code> and <code>C</code> are all <code>Ring</code>s when taken individually</li>
<li><code>to&lt;C&gt; : A -&gt; B</code> and <code>to&lt;C&gt; : B -&gt; C</code> are <code>Ring</code>-embeddings, as determined by the <code>is_embedding</code> metafunction.</li>
</ol>
<p>The definition of <code>mult</code> for data types satisfying the above properties is obtained by setting </p><div class="fragment"><div class="line"><a class="code" href="group__group-_ring.html#ga052d31c269a6a438cc8004c9ad1efdfa">mult</a>(x, y) = <a class="code" href="group__group-_ring.html#ga052d31c269a6a438cc8004c9ad1efdfa">mult</a>(to&lt;C&gt;(x), to&lt;C&gt;(y))</div>
</div><!-- fragment --><h2><a class="anchor" id="autotoc_md287"></a>
Example</h2>
<div class="fragment"><div class="line"><span class="comment">// Copyright Louis Dionne 2013-2017</span></div>
<div class="line"><span class="comment">// Distributed under the Boost Software License, Version 1.0.</span></div>
<div class="line"><span class="comment">// (See accompanying file LICENSE.md or copy at http://boost.org/LICENSE_1_0.txt)</span></div>
<div class="line"> </div>
<div class="line"><span class="preprocessor">#include &lt;<a class="code" href="assert_8hpp.html">boost/hana/assert.hpp</a>&gt;</span></div>
<div class="line"><span class="preprocessor">#include &lt;<a class="code" href="equal_8hpp.html">boost/hana/equal.hpp</a>&gt;</span></div>
<div class="line"><span class="preprocessor">#include &lt;<a class="code" href="integral__constant_8hpp.html">boost/hana/integral_constant.hpp</a>&gt;</span></div>
<div class="line"><span class="preprocessor">#include &lt;<a class="code" href="mult_8hpp.html">boost/hana/mult.hpp</a>&gt;</span></div>
<div class="line"><span class="keyword">namespace </span>hana = <a class="code" href="namespaceboost_1_1hana.html">boost::hana</a>;</div>
<div class="line"> </div>
<div class="line"> </div>
<div class="line"><a class="code" href="group__group-assertions.html#ga2c5006540936d9f8880e3a39f4fcc035">BOOST_HANA_CONSTANT_CHECK</a>(<a class="code" href="group__group-_ring.html#ga052d31c269a6a438cc8004c9ad1efdfa">hana::mult</a>(hana::int_c&lt;3&gt;, hana::int_c&lt;5&gt;) == hana::int_c&lt;15&gt;);</div>
<div class="line">static_assert(<a class="code" href="group__group-_ring.html#ga052d31c269a6a438cc8004c9ad1efdfa">hana::mult</a>(4, 2) == 8, <span class="stringliteral">&quot;&quot;</span>);</div>
<div class="line"> </div>
<div class="line"><span class="keywordtype">int</span> main() { }</div>
<div class="ttc" id="aassert_8hpp_html"><div class="ttname"><a href="assert_8hpp.html">assert.hpp</a></div><div class="ttdoc">Defines macros to perform different kinds of assertions.</div></div>
<div class="ttc" id="aequal_8hpp_html"><div class="ttname"><a href="equal_8hpp.html">equal.hpp</a></div><div class="ttdoc">Defines boost::hana::equal.</div></div>
<div class="ttc" id="agroup__group-assertions_html_ga2c5006540936d9f8880e3a39f4fcc035"><div class="ttname"><a href="group__group-assertions.html#ga2c5006540936d9f8880e3a39f4fcc035">BOOST_HANA_CONSTANT_CHECK</a></div><div class="ttdeci">#define BOOST_HANA_CONSTANT_CHECK(...)</div><div class="ttdoc">Equivalent to BOOST_HANA_CONSTANT_ASSERT, but not influenced by the BOOST_HANA_CONFIG_DISABLE_ASSERTI...</div><div class="ttdef"><b>Definition:</b> assert.hpp:239</div></div>
<div class="ttc" id="aintegral__constant_8hpp_html"><div class="ttname"><a href="integral__constant_8hpp.html">integral_constant.hpp</a></div><div class="ttdoc">Defines boost::hana::integral_constant.</div></div>
<div class="ttc" id="amult_8hpp_html"><div class="ttname"><a href="mult_8hpp.html">mult.hpp</a></div><div class="ttdoc">Defines boost::hana::mult.</div></div>
<div class="ttc" id="anamespaceboost_1_1hana_html"><div class="ttname"><a href="namespaceboost_1_1hana.html">boost::hana</a></div><div class="ttdoc">Namespace containing everything in the library.</div><div class="ttdef"><b>Definition:</b> accessors.hpp:20</div></div>
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<a id="gadea531feb3b0a1c5c3d777f7ab45e932"></a>
<h2 class="memtitle"><span class="permalink"><a href="#gadea531feb3b0a1c5c3d777f7ab45e932">&#9670;&nbsp;</a></span>one</h2>

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<p><code>#include &lt;<a class="el" href="fwd_2one_8hpp.html">boost/hana/fwd/one.hpp</a>&gt;</code></p>
<b>Initial value:</b><div class="fragment"><div class="line">= []() -&gt; decltype(<span class="keyword">auto</span>) {</div>
<div class="line">        <span class="keywordflow">return</span> tag-dispatched;</div>
<div class="line">    }</div>
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<p>Identity of the <code>Ring</code> multiplication. </p>
<dl class="tparams"><dt>Template Parameters</dt><dd>
  <table class="tparams">
    <tr><td class="paramname">R</td><td>The tag (must be a model of <code>Ring</code>) of the returned identity.</td></tr>
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Example</h2>
<div class="fragment"><div class="line"><span class="comment">// Copyright Louis Dionne 2013-2017</span></div>
<div class="line"><span class="comment">// Distributed under the Boost Software License, Version 1.0.</span></div>
<div class="line"><span class="comment">// (See accompanying file LICENSE.md or copy at http://boost.org/LICENSE_1_0.txt)</span></div>
<div class="line"> </div>
<div class="line"><span class="preprocessor">#include &lt;<a class="code" href="assert_8hpp.html">boost/hana/assert.hpp</a>&gt;</span></div>
<div class="line"><span class="preprocessor">#include &lt;<a class="code" href="equal_8hpp.html">boost/hana/equal.hpp</a>&gt;</span></div>
<div class="line"><span class="preprocessor">#include &lt;<a class="code" href="integral__constant_8hpp.html">boost/hana/integral_constant.hpp</a>&gt;</span></div>
<div class="line"><span class="preprocessor">#include &lt;<a class="code" href="one_8hpp.html">boost/hana/one.hpp</a>&gt;</span></div>
<div class="line"><span class="keyword">namespace </span>hana = <a class="code" href="namespaceboost_1_1hana.html">boost::hana</a>;</div>
<div class="line"> </div>
<div class="line"> </div>
<div class="line"><a class="code" href="group__group-assertions.html#ga2c5006540936d9f8880e3a39f4fcc035">BOOST_HANA_CONSTANT_CHECK</a>(<a class="code" href="group__group-_ring.html#gadea531feb3b0a1c5c3d777f7ab45e932">hana::one</a>&lt;hana::integral_constant_tag&lt;int&gt;&gt;() == hana::int_c&lt;1&gt;);</div>
<div class="line">static_assert(hana::one&lt;long&gt;() == 1l, <span class="stringliteral">&quot;&quot;</span>);</div>
<div class="line"> </div>
<div class="line"><span class="keywordtype">int</span> main() { }</div>
<div class="ttc" id="agroup__group-_ring_html_gadea531feb3b0a1c5c3d777f7ab45e932"><div class="ttname"><a href="group__group-_ring.html#gadea531feb3b0a1c5c3d777f7ab45e932">boost::hana::one</a></div><div class="ttdeci">constexpr auto one</div><div class="ttdoc">Identity of the Ring multiplication.</div><div class="ttdef"><b>Definition:</b> one.hpp:30</div></div>
<div class="ttc" id="aone_8hpp_html"><div class="ttname"><a href="one_8hpp.html">one.hpp</a></div><div class="ttdoc">Defines boost::hana::one.</div></div>
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<a id="ga0ee3cff9ec646bcc7217f00ee6099b72"></a>
<h2 class="memtitle"><span class="permalink"><a href="#ga0ee3cff9ec646bcc7217f00ee6099b72">&#9670;&nbsp;</a></span>power</h2>

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          <td class="memname">constexpr auto boost::hana::power</td>
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  <td class="mlabels-right">
<span class="mlabels"><span class="mlabel">constexpr</span></span>  </td>
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<p><code>#include &lt;<a class="el" href="fwd_2power_8hpp.html">boost/hana/fwd/power.hpp</a>&gt;</code></p>
<b>Initial value:</b><div class="fragment"><div class="line">= [](<span class="keyword">auto</span>&amp;&amp; x, <span class="keyword">auto</span> <span class="keyword">const</span>&amp; n) -&gt; decltype(<span class="keyword">auto</span>) {</div>
<div class="line">        <span class="keywordflow">return</span> tag-dispatched;</div>
<div class="line">    }</div>
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<p>Elevate a ring element to its <code>n</code>th power. </p>
<p>Specifically, <code>power(x, n)</code>, is equivalent to multiplying <code>x</code> with itself <code>n</code> times using the Ring's multiplication. If the power is equal to <code>zero</code>, the Ring's identity (<code>one</code>) is returned.</p>
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">x</td><td>A <code>Ring</code> element that is elevated to its <code>n</code>th power.</td></tr>
    <tr><td class="paramname">n</td><td>A non-negative <code><a class="el" href="structboost_1_1hana_1_1_integral_constant.html" title="The IntegralConstant concept represents compile-time integral values.">IntegralConstant</a></code> representing the power to which <code>x</code> is elevated.</td></tr>
  </table>
  </dd>
</dl>
<dl class="section note"><dt>Note</dt><dd>Only the tag of <code>x</code> is used for tag-dispatching.</dd></dl>
<h2><a class="anchor" id="autotoc_md319"></a>
Example</h2>
<div class="fragment"><div class="line"><span class="comment">// Copyright Louis Dionne 2013-2017</span></div>
<div class="line"><span class="comment">// Distributed under the Boost Software License, Version 1.0.</span></div>
<div class="line"><span class="comment">// (See accompanying file LICENSE.md or copy at http://boost.org/LICENSE_1_0.txt)</span></div>
<div class="line"> </div>
<div class="line"><span class="preprocessor">#include &lt;<a class="code" href="assert_8hpp.html">boost/hana/assert.hpp</a>&gt;</span></div>
<div class="line"><span class="preprocessor">#include &lt;<a class="code" href="equal_8hpp.html">boost/hana/equal.hpp</a>&gt;</span></div>
<div class="line"><span class="preprocessor">#include &lt;<a class="code" href="integral__constant_8hpp.html">boost/hana/integral_constant.hpp</a>&gt;</span></div>
<div class="line"><span class="preprocessor">#include &lt;<a class="code" href="power_8hpp.html">boost/hana/power.hpp</a>&gt;</span></div>
<div class="line"><span class="keyword">namespace </span>hana = <a class="code" href="namespaceboost_1_1hana.html">boost::hana</a>;</div>
<div class="line"> </div>
<div class="line"> </div>
<div class="line"><a class="code" href="group__group-assertions.html#ga2c5006540936d9f8880e3a39f4fcc035">BOOST_HANA_CONSTANT_CHECK</a>(<a class="code" href="group__group-_ring.html#ga0ee3cff9ec646bcc7217f00ee6099b72">hana::power</a>(hana::int_c&lt;3&gt;, hana::int_c&lt;2&gt;) == hana::int_c&lt;3 * 3&gt;);</div>
<div class="line">static_assert(<a class="code" href="group__group-_ring.html#ga0ee3cff9ec646bcc7217f00ee6099b72">hana::power</a>(2, hana::int_c&lt;4&gt;) == 16, <span class="stringliteral">&quot;&quot;</span>);</div>
<div class="line"> </div>
<div class="line"><span class="keywordtype">int</span> main() { }</div>
<div class="ttc" id="agroup__group-_ring_html_ga0ee3cff9ec646bcc7217f00ee6099b72"><div class="ttname"><a href="group__group-_ring.html#ga0ee3cff9ec646bcc7217f00ee6099b72">boost::hana::power</a></div><div class="ttdeci">constexpr auto power</div><div class="ttdoc">Elevate a ring element to its nth power.</div><div class="ttdef"><b>Definition:</b> power.hpp:40</div></div>
<div class="ttc" id="apower_8hpp_html"><div class="ttname"><a href="power_8hpp.html">power.hpp</a></div><div class="ttdoc">Defines boost::hana::power.</div></div>
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